Summary of discussion on the afternoon of 4 April 2004.

Ron Goldman:
  Geometric modeling, even though it deals with algebraic objects arose
  out of numerical analysis.  It is an intrinsically numerical (floating-point)
  subject.  What Geometric Modelling needs is "Numerical Algebraic Geometry".

  Example of a research question:  What is the best (Numerically stable, robust,
   quickest, etc.) method to compute the resultant of two polynomials?

Sottile:  There has been some work in numerical algebraic geometry.  Stetter
  and others tried to come up with a theory of numerical Groebner bases.
  Verschelde, Sommese, and Wampler, are using numerical homotopy methods to 
  establish a `numerical algebraic geometry'.  Smale and his collaborators 
  also have a school in these directions.

  Useful research direction: Implementation of a good exclusion method for 
   finding real solutions to polynomial systems.  Interesting progress 
   likely using a mixture of exclusion methods and Ming Zhang's Bezier-subdivision.

  Another one:  Approximate implicitization?   This has been problematic in the 
   past.

  Schicho asks: Given a collection of polynomials that parametrize a patch, we can compute 
   an implicit equation.  How (i.e. continuity, sensitivity to inputs) does 
   the implicit equation depend upon the input polynomials.

   How about curves?

  Rimas:   Currently, structural issues are not well-understood.  We need to understand possible
   structures of patches and then develop them.  For example, is it possible to axiomitize 
   M-patches?  Can we prove Theorems like, if A holds, then we have Blossoming?
 
   Can we prove that a class of example has
   1) uniqueness from the input data.
   2) A universal rational parametrization, or is universal in some sense.
   3) Has parametrizations of minimal degree.

   How do base points influence the shapes of M-patches?

   How about the term `almost toric surface' to mean a linear projection of a normal toric 
    surface.

   For a rational parametrization.  How to control its features, e.g. speed?  (Metric
    properties of a parametrization.)

   Are there other natural rational varieties that may be useful for geometric modeling?

   (Sturmfels suggests Grassmannians, Schubert varieties, determinantal varieties, 
    or spherical varieties.)
    What are the interesting 2-dimensional varieties in these families?

   Given Bezier curves on a rational surface (implicit or parametrized) that bound a polygon.
    How to find a parametrization to:
     1) fill in the polygonal hole
     2) preserve the parametrization of the curves (if possible)
     3) Is there an obstruction theory to this?  I.e homotopy theory, or global extension 
        of a section?  (It appears to be so to Frank.)

   Is geometric modeling useful for other problems, for example to higher-dimensional modelling.

   What can be said about linear precision?
    -> What shapes have rational parametrizations with linear precision.

   For Rational Patches (e.g Patches a' la Be'zier), can we have good control of the 
    boundary curves (e.g. Bezier curves).   How about linear precision on the boundary, 
    or C^1 (or higher) conditioning near the boundary.

   What is the relation between the Cox ring of a del Pezzo surface and rational parametrizations 
    of it?

   Is CAGD in a crisis?  
    -> Main questions have been solved, and there may be a shortage of new ideas?

-----------------------------------------
 Possible future directions:

   Possible collaborative grants, maybe centered on texas
    Something to support collaboration, students, travel, conferences.
    International component.  Links with the European project in 
    this topic.

 --> Talk to both CS and Math NSF program officers about this possibility.

  There is a definite interest in both sides, but there is a need to train
   students and postdocs to cary these ideas forward.

  Links with people from the video gaming industry.  E.G. Pixar.